Lorenz curve

The Lorenz curve was developed by Max O. Lorenz as a graphical representation of income inequality. It can also be used to measure inequality of assets or other distributions.

In discussions of personal income, we frequently make statements such as, "the bottom twenty percent of all households have ten percent of the total income". The Lorenz curve is based on such statements; every point on the curve represents one such statement.

The Lorenz curve is a graph that shows, for the bottom x% of households, the percentage y% of the total income which they have. The percentage of households is plotted on the x axis, the percentage of income on the y axis.

A perfectly equal income distribution in a society would be one in which every person has the same income. In this case, the bottom N% of society would always have N% of the income. Thus a perfectly equal distribution can be depicted by the straight line y=x; we call this line the line of perfect equality.

A perfectly inequal distribution, by contrast, would be one in which one person has all the income and everyone else has none. In that case, the curve would be at y=0 for all x<100, and y=100 when x=100. We call this curve the line of perfect inequality.

Note that it is impossible for the Lorenz curve to rise above the line of perfect equality, or sink below the line of perfect inequality.

The Lorenz curve is used to calculate the Gini coefficient, which is the area between the line of perfect equality and the Lorenz curve, as a percentage of the area between the line of perfect equality and the line of perfect inequality.

The Lorenz curve looks like this:

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